Three-level saturated orthogonal arrays with less β-wordlength pattern

Saturated orthogonal array is an important type of fractional factorial designs in experimental design. On the basis of guaranteeing orthogonality, it maximizes the number of factors and saves the cost of experiments. Therefore, it is increasingly used in practical experiment. For quantitative factors, the minimum β-aberration criterion is suitable for selecting good designs under a polynomial model. However, construction methods of designs with less β-aberration have not been fully investigated. In this article, we apply the Rao-Hamming method to construct a class of three-level saturated orthogonal arrays, and perform a special level permutation on the designs of the Rao-Hamming construction, which makes the resultant designs mirror-symmetric ones. Furthermore, we analyze the distribution of elements in each row and among all possible pairs of rows of the mirror-symmetric designs, and discuss some properties of the β-wordlength pattern for three-level mirror-symmetric designs constructed by the Rao-Hamming method. In addition, we provide the explicit expressions of β3 and β4 for three-level mirror-symmetric designs, which can reduce the computational complexity of the β-wordlength pattern, and provide an effective idea for finding designs under the minimum β-aberration criterion. Finally, numerical examples are used to illustrate and verify the results.